OFFSET
1,1
COMMENTS
A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.
Also Heinz numbers of non-weakly alternating non-strict integer partitions, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. These partitions are counted by A349796. This sequence involves the somewhat degenerate case where no strict increases are allowed.
EXAMPLE
The terms together with their Heinz partitions begin (A-E = 10-14):
60: (3211) 276: (9211) 420: (43211)
84: (4211) 280: (43111) 440: (53111)
120: (32111) 294: (4421) 444: (C211)
132: (5211) 300: (33211) 456: (82111)
140: (4311) 308: (5411) 460: (9311)
150: (3321) 312: (62111) 476: (7411)
156: (6211) 315: (4322) 480: (3211111)
168: (42111) 336: (421111) 490: (4431)
204: (7211) 340: (7311) 492: (D211)
220: (5311) 348: (A211) 495: (5322)
228: (8211) 364: (6411) 516: (E211)
240: (321111) 372: (B211) 520: (63111)
260: (6311) 378: (42221) 528: (521111)
264: (52111) 380: (8311) 532: (8411)
270: (32221) 408: (72111) 540: (322211)
MATHEMATICA
Select[Range[300], !SquareFreeQ[#]&&PrimeNu[#]>1&& !And@@EvenQ/@Take[Last/@FactorInteger[#], {2, -2}]&]
CROSSREFS
These partitions are counted by A349796.
A003242 = Carlitz (anti-run) compositions.
A096441 = weakly alternating 0-appended partitions.
A349056 = weakly alternating permutations of prime indices.
A349798 = weakly but not strongly alternating perms of prime indices.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 25 2021
STATUS
approved